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Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the
infix An infix is an affix inserted inside a word stem (an existing word or the core of a family of words). It contrasts with '' adfix,'' a rare term for an affix attached to the outside of a stem, such as a prefix or suffix. When marking text for i ...
operators XOR ( or ), EOR, EXOR, , , , , \nleftrightarrow, and . The
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
of XOR is the logical biconditional, which yields true if and only if the two inputs are the same. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both
operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above exam ...
s are true; the exclusive or operator ''excludes'' that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B". Since it is associative, it may be considered to be an ''n''-ary operator which is true if and only if an odd number of arguments are true. That is, ''a'' XOR ''b'' XOR ... may be treated as XOR(''a'',''b'',...).


Truth table

The
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
of A XOR B shows that it outputs true whenever the inputs differ:


Equivalences, elimination, and introduction

Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction p \nleftrightarrow q, also denoted by p ? q or \operatornamepq, can be expressed in terms of the
logical conjunction In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...
("logical and", \wedge), the
disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...
("logical or", \lor), and the
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
(\lnot) as follows: : \begin p \nleftrightarrow q & = & (p \lor q) \land \lnot (p \land q) \end The exclusive disjunction p \nleftrightarrow q can also be expressed in the following way: : \begin p \nleftrightarrow q & = & (p \land \lnot q) \lor (\lnot p \land q) \end This representation of XOR may be found useful when constructing a circuit or network, because it has only one \lnot operation and small number of \wedge and \lor operations. A proof of this identity is given below: : \begin p \nleftrightarrow q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ pt & = & ((p \land \lnot q) \lor \lnot p) & \land & ((p \land \lnot q) \lor q) \\ pt & = & ((p \lor \lnot p) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\ pt & = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\ pt & = & \lnot (p \land q) & \land & (p \lor q) \end It is sometimes useful to write p \nleftrightarrow q in the following way: : \begin p \nleftrightarrow q & = & \lnot ((p \land q) \lor (\lnot p \land \lnot q)) \end or: : \begin p \nleftrightarrow q & = & (p \lor q) \land (\lnot p \lor \lnot q) \end This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof. The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication (a material conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and
material equivalence Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geologic ...
. In summary, we have, in mathematical and in engineering notation: : \begin p \nleftrightarrow q & = & (p \land \lnot q) & \lor & (\lnot p \land q) & = & p\overline + \overlineq \\ pt & = & (p \lor q) & \land & (\lnot p \lor \lnot q) & = & (p + q)(\overline + \overline) \\ pt & = & (p \lor q) & \land & \lnot (p \land q) & = & (p + q)(\overline) \end


Negation

The spirit of De Morgan's laws can be applied, we have: \lnot(p \nleftrightarrow q) = \lnot p \nleftrightarrow q = p \nleftrightarrow \lnot q


Relation to modern algebra

Although the operators \wedge ( conjunction) and \lor (
disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...
) are very useful in logic systems, they fail a more generalizable structure in the following way: The systems (\, \wedge) and (\, \lor) are
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
s, but neither is a group. This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring. However, the system using exclusive or (\, \oplus) ''is'' an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. The combination of operators \wedge and \oplus over elements \ produce the well-known field F_2. This field can represent any logic obtainable with the system (\land, \lor) and has the added benefit of the arsenal of algebraic analysis tools for fields. More specifically, if one associates F with 0 and T with 1, one can interpret the logical "AND" operation as multiplication on F_2 and the "XOR" operation as addition on F_2: :\begin r = p \land q & \Leftrightarrow & r = p \cdot q \pmod 2 \\ pt r = p \oplus q & \Leftrightarrow & r = p + q \pmod 2 \\ \end Using this basis to describe a boolean system is referred to as
algebraic normal form In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), '' Zhegalkin normal form'', or ''Reed–Muller expansion'' is a way of writing logical formulas in one of three subforms: * The entire formula is purely tru ...
.


Exclusive or in natural language

Disjunction is often understood exclusively in
natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
s. In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet. :1. Mary is a singer or a poet. However, disjunction can also be understood inclusively, even in combination with "either". For instance, the first example below shows that "either" can be felicitously used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under downward entailing contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans. :2. Mary is either a singer or a poet or both. :3. Nobody ate either rice or beans. Examples such as the above have motivated analyses of the exclusivity inference as
pragmatic Pragmatism is a philosophical movement. Pragmatism or pragmatic may also refer to: *Pragmaticism, Charles Sanders Peirce's post-1905 branch of philosophy * Pragmatics, a subfield of linguistics and semiotics *'' Pragmatics'', an academic journal i ...
conversational implicature In pragmatics, a subdiscipline of linguistics, an implicature is something the speaker suggests or implies with an utterance, even though it is not literally expressed. Implicatures can aid in communicating more efficiently than by explicitly sayi ...
s calculated on the basis of an inclusive
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
. Implicatures are typically cancellable and do not arise in downward entailing contexts if their calculation depends on the Maxim of Quantity. However, some researchers have treated exclusivity as a bona fide semantic entailment and proposed nonclassical logics which would validate it. This behavior of English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French ''soit... soit''.


Alternative symbols

The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen: * +, a plus sign, which has the advantage that all of the ordinary algebraic properties of mathematical rings and fields can be used without further ado; but the plus sign is also used for inclusive disjunction in some notation systems; note that exclusive disjunction corresponds to
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
2, which has the following addition table, clearly
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the one above: * \oplus, a modified plus sign; this symbol is also used in mathematics for the ''
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
'' of algebraic structures * J, as in J''pq'' * An inclusive disjunction symbol (\lor) that is modified in some way, such as ** \underline\lor ** \dot\vee * ^, the caret, used in several
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s, such as C, C++, C#, D,
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
,
Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offic ...
,
Ruby A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum ( aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called ...
, PHP and Python, denoting the
bitwise In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
XOR operator; not used outside of programming contexts because it is too easily confused with other uses of the caret such as exponentiation. * , sometimes written as ** >< ** >-< * =1, in IEC symbology


Properties

If using binary values for true (1) and false (0), then ''exclusive or'' works exactly like
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
2.


Computer science


Bitwise operation

Exclusive disjunction is often used for bitwise operations. Examples: * 1 XOR 1 = 0 * 1 XOR 0 = 1 * 0 XOR 1 = 1 * 0 XOR 0 = 0 * XOR = (this is equivalent to addition without carry) As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two ''n''-bit strings is identical to the standard vector of addition in the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(\Z/2\Z)^n. In computer science, exclusive disjunction has several uses: * It tells whether two bits are unequal. * It is an optional bit-flipper (the deciding input chooses whether to invert the data input). * It tells whether there is an
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
number of 1 bits (A \oplus B \oplus C \oplus D \oplus E is true
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
an odd number of the variables are true), which is equal to the
parity bit A parity bit, or check bit, is a bit added to a string of binary code. Parity bits are a simple form of error detecting code. Parity bits are generally applied to the smallest units of a communication protocol, typically 8-bit octets (bytes), ...
returned by a parity function. In logical circuits, a simple adder can be made with an XOR gate to add the numbers, and a series of AND, OR and NOT gates to create the carry output. On some computer architectures, it is more efficient to store a zero in a register by XOR-ing the register with itself (bits XOR-ed with themselves are always zero) instead of loading and storing the value zero. In simple threshold-activated
neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
s, modeling the XOR function requires a second layer because XOR is not a linearly separable function. Exclusive-or is sometimes used as a simple mixing function in
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adv ...
, for example, with
one-time pad In cryptography, the one-time pad (OTP) is an encryption technique that cannot be cracked, but requires the use of a single-use pre-shared key that is not smaller than the message being sent. In this technique, a plaintext is paired with a ra ...
or Feistel network systems. Exclusive-or is also heavily used in block ciphers such as AES (Rijndael) or Serpent and in block cipher implementation (CBC, CFB, OFB or CTR). Similarly, XOR can be used in generating entropy pools for hardware random number generators. The XOR operation preserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source. XOR is used in RAID 3–6 for creating parity information. For example, RAID can "back up" bytes and from two (or more) hard drives by XORing the just mentioned bytes, resulting in () and writing it to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. For instance, if the drive containing is lost, and can be XORed to recover the lost byte. XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "1" if there is an overflow. XOR can be used to swap two numeric variables in computers, using the XOR swap algorithm; however this is regarded as more of a curiosity and not encouraged in practice. XOR linked lists leverage XOR properties in order to save space to represent doubly linked list data structures. In
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, XOR-based drawing methods are often used to manage such items as bounding boxes and cursors on systems without
alpha channels In computer graphics, alpha compositing or alpha blending is the process of combining one image with a background to create the appearance of partial or full transparency. It is often useful to render picture elements (pixels) in separate pas ...
or overlay planes.


Encodings

It is also called "not left-right arrow" (\nleftrightarrow) in
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
-based markdown (\nleftrightarrow). Apart from the ASCII codes, the operator is encoded at and , both in block
mathematical operators Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation. Notably absent are the plus sign (+), greater than sign (>) and less than sign (<), due to them already appearing in the Bas ...
.


See also


Notes


External links


All About XOR

Proofs of XOR properties and applications of XOR, CS103: Mathematical Foundations of Computing, Stanford University
{{Logical connectives Dichotomies Logical connectives Semantics